x+1/x^2+1-1/2=0 eching

$$\frac{ x+1 }{ { x }^{ 2 } +1 } - \frac{ 1 }{ 2 } =0$$

x uchun yechish

$$x=\sqrt{2}+1\approx 2,414213562$$

$$x=1-\sqrt{2}\approx -0,414213562$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 2\left(x^{2}+1\right) ga, x^{2}+1,2 ning eng kichik karralisiga ko‘paytiring.

$$2\left(x+1\right)+2\left(x^{2}+1\right)\left(-\frac{1}{2}\right)=0$$

2 ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x+2+2\left(x^{2}+1\right)\left(-\frac{1}{2}\right)=0$$

-1 hosil qilish uchun 2 va -\frac{1}{2} ni ko'paytirish.

$$2x+2-\left(x^{2}+1\right)=0$$

x^{2}+1 teskarisini topish uchun har birining teskarisini toping.

$$2x+2-x^{2}-1=0$$

1 olish uchun 2 dan 1 ni ayirish.

$$2x+1-x^{2}=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$-x^{2}+2x+1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 2 ni b va 1 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)}}{2\left(-1\right)}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+4}}{2\left(-1\right)}$$

4 ni 4 ga qo'shish.

$$x=\frac{-2±\sqrt{8}}{2\left(-1\right)}$$

8 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{2}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-2±2\sqrt{2}}{-2}$$

x=\frac{-2±2\sqrt{2}}{-2} tenglamasini yeching, bunda ± musbat. -2 ni 2\sqrt{2} ga qo'shish.

$$x=\frac{2\sqrt{2}-2}{-2}$$

-2+2\sqrt{2} ni -2 ga bo'lish.

$$x=1-\sqrt{2}$$

x=\frac{-2±2\sqrt{2}}{-2} tenglamasini yeching, bunda ± manfiy. -2 dan 2\sqrt{2} ni ayirish.

$$x=\frac{-2\sqrt{2}-2}{-2}$$

-2-2\sqrt{2} ni -2 ga bo'lish.

$$x=\sqrt{2}+1$$

Tenglama yechildi.

$$x=1-\sqrt{2} x=\sqrt{2}+1$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 2\left(x^{2}+1\right) ga, x^{2}+1,2 ning eng kichik karralisiga ko‘paytiring.

$$2\left(x+1\right)+2\left(x^{2}+1\right)\left(-\frac{1}{2}\right)=0$$

2 ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x+2+2\left(x^{2}+1\right)\left(-\frac{1}{2}\right)=0$$

-1 hosil qilish uchun 2 va -\frac{1}{2} ni ko'paytirish.

$$2x+2-\left(x^{2}+1\right)=0$$

x^{2}+1 teskarisini topish uchun har birining teskarisini toping.

$$2x+2-x^{2}-1=0$$

1 olish uchun 2 dan 1 ni ayirish.

$$2x+1-x^{2}=0$$

Ikkala tarafdan 1 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$2x-x^{2}=-1$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-x^{2}+2x=-1$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+2x}{-1}=-\frac{1}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{2}{-1}x=-\frac{1}{-1}$$

2 ni -1 ga bo'lish.

$$x^{2}-2x=-\frac{1}{-1}$$

-1 ni -1 ga bo'lish.

$$x^{2}-2x=1$$

-2 ni bo‘lish, x shartining koeffitsienti, 2 ga -1 olish uchun. Keyin, -1 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-2x+1=1+1$$

1 ni 1 ga qo'shish.

$$x^{2}-2x+1=2$$

x^{2}-2x+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-1\right)^{2}=2$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{2}$$

Qisqartirish.

$$x-1=\sqrt{2} x-1=-\sqrt{2}$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\sqrt{2}+1 x=1-\sqrt{2}$$